cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A058899 Inconsummate numbers in base 3: no number is this multiple of the sum of its digits (in base 3).

Original entry on oeis.org

17, 32, 44, 51, 94, 95, 96, 106, 107, 112, 118, 132, 148, 153, 199, 224, 226, 232, 235, 236, 238, 256, 265, 268, 269, 274, 277, 281, 282, 284, 285, 288, 296, 308, 318, 321, 334, 336, 343, 352, 354, 368, 396, 442, 443, 444, 454, 459, 469, 472
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Links

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Mathematica
    base=3; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250 n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Sep 23 2017 *)
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058899_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 2*l*n < 3**(l-1):
                yield n
                break
            for d in combinations_with_replacement((0,1,2),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,3)[1:]) == list(d):
                    break
            else:
                continue
            break
A058899_list = list(islice(A058899_gen(),20)) # Chai Wah Wu, May 10 2023

A342593 Numbers m that are not the quotient of a Zuckerman number divided by the product of its digits.

Original entry on oeis.org

10, 15, 16, 20, 24, 25, 26, 30, 32, 35, 38, 39, 40, 42, 43, 47, 50, 54, 55, 58, 60, 62, 65, 70, 71, 73, 75, 78, 80, 85, 87, 90, 92, 95, 99, 100, 105, 107, 108, 110, 115, 116, 117, 119, 120, 123, 125, 127, 130, 131, 135, 137, 138, 139, 140, 141, 142, 145, 146, 147, 150, 155
Offset: 1

Views

Author

Bernard Schott, Mar 16 2021

Keywords

Comments

The Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits (see link).
m is a term iff A056770(m) = 0.
All the multiples of 10 are terms.
Many numbers that end with 5 are terms, first exceptions < 1000: 5, 45, 255, 315, 505, ...

Links

Crossrefs

Cf. A007602, A056770, A288069.
Cf. A003635 (similar for Niven numbers).

A058900 Inconsummate numbers in base 4: no number is this multiple of the sum of its digits (in base 4).

Original entry on oeis.org

29, 41, 71, 83, 93, 111, 113, 114, 116, 117, 122, 123, 125, 135, 137, 143, 146, 153, 164, 167, 191, 197, 201, 237, 242, 263, 275, 279, 282, 284, 285, 291, 303, 305, 311, 323, 326, 327, 332, 359, 362, 369, 372, 375, 377, 382, 383, 389, 407, 410
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Links

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058900_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 3*l*n < 1<<((l-1)<<1):
                yield n
                break
            for d in combinations_with_replacement((0,1,2,3),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,4)[1:]) == list(d):
                    break
            else:
                continue
            break
A058900_list = list(islice(A058900_gen(),20)) # Chai Wah Wu, May 10 2023

A058901 Inconsummate numbers in base 5: no number is this multiple of the sum of its digits (in base 5).

Original entry on oeis.org

16, 22, 28, 46, 56, 58, 68, 74, 76, 80, 106, 108, 110, 118, 128, 136, 138, 140, 146, 152, 168, 198, 202, 206, 208, 230, 249, 256, 258, 262, 263, 268, 274, 276, 278, 280, 284, 286, 288, 290, 292, 294, 296, 298, 302, 318, 323, 324, 326, 336, 338
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Links

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Mathematica
    base=5; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^4}] (* Vincenzo Librandi, Nov 03 2016 *)
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058901_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 4*l*n < 5**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(5),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,5)[1:]) == list(d):
                    break
            else:
                continue
            break
A058901_list = list(islice(A058901_gen(),20)) # Chai Wah Wu, May 10 2023

A130338 Primes p with no solution x to x=p*digitsum(x).

Original entry on oeis.org

173, 383, 431, 443, 461, 491, 521, 563, 761, 821, 827, 839, 941, 971, 983, 1049, 1481, 1487, 1493, 1499, 1553, 1571, 1601, 1811, 1871, 1931, 2153, 2207, 2477, 2591, 2609, 2753, 3037, 3041, 3083, 3137, 3221, 3251, 3257, 3307, 3329, 3371
Offset: 1

Views

Author

Lekraj Beedassy, Aug 07 2007

Keywords

Comments

Primes p such that no number is p times its digit sum.
These may be called the non-Moran primes because no index k exists in A001101 to represent them as A001101(k)/digitsum[A001101(k)]. - R. J. Mathar, Aug 10 2007

Examples

			p=5743 is not in the sequence because it can be represented as p=40201/7 (x=40201) or as p=80402/14 (x=80402).
p=7 is not in the sequence because it can be represented as p=21/3 (x=21) or p=42/6 (x=42) or p=63/9 (x=63) or p=84/12 (x=84). In all cases, the denominators are the digit sums of the numerators.

Links

Crossrefs

Cf. A003635.
Cf. A000040, A001101, A007953, A003635, A066007.

Programs

  • Maple
    A007953 := proc(n) option remember ; add(j,j=convert(n,base,10)) ; end: A001101 := proc(p) option remember : local k,digs ; digs := 1; if not isprime(p) then RETURN(-1) ; else while 10^(digs-1)/(9*digs) <= p do for k from max(p,10^(digs-1)) to 10^digs do if k = p*A007953(k) then RETURN(k) ; fi ; od ; digs := digs+1 ; od: RETURN(-1) ; fi ; end: for n from 1 to 500 do if A001101(ithprime(n)) = -1 then printf("%d,",ithprime(n)) ; fi : od: # R. J. Mathar, Aug 10 2007
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy import nextprime
def A130338_gen(startvalue=1): # generator of terms >= startvalue
    n = nextprime(max(startvalue,1)-1)
    while True:
        for l in count(1):
            if 9*l*n < 10**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(10),l):
                if (s:=sum(d))>0 and sorted(str(s*n)) == [str(e) for e in d]:
                    break
            else:
                continue
            break
        n = nextprime(n)
A130338_list = list(islice(A130338_gen(),20)) # Chai Wah Wu, May 09 2023

Formula

A000040 MINUS {A001101(k)/A007953(A001101(k)): k=1,2,3,4,..}. A003635 INTERSECT A000040. - R. J. Mathar, Aug 10 2007

Extensions

More terms from R. J. Mathar, Aug 10 2007

A058902 Inconsummate numbers in base 6: no number is this multiple of the sum of its digits (in base 6).

Original entry on oeis.org

27, 33, 64, 82, 97, 100, 103, 104, 107, 118, 122, 124, 125, 128, 134, 135, 152, 159, 162, 177, 190, 193, 195, 198, 205, 208, 212, 214, 232, 233, 250, 277, 280, 298, 334, 343, 345, 349, 352, 358, 362, 363, 364, 370, 380, 382, 384, 403, 427, 442
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Links

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Mathematica
    base=6; Do[k=n; While[Apply[Plus,IntegerDigits[k, base]] n!=k&&k<250 n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Jan 30 2017 *)
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058902_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 5*l*n < 6**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(6),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,6)[1:]) == list(d):
                    break
            else:
                continue
            break
A058902_list = list(islice(A058902_gen(),20)) # Chai Wah Wu, May 10 2023

A058903 Inconsummate numbers in base 7: no number is this multiple of the sum of its digits (in base 7).

Original entry on oeis.org

30, 86, 102, 134, 138, 141, 158, 162, 167, 170, 183, 186, 194, 210, 213, 233, 284, 290, 306, 312, 314, 326, 330, 338, 354, 362, 366, 368, 428, 452, 480, 530, 534, 536, 540, 542, 554, 564, 578, 591, 602, 645, 648, 656, 705, 708, 714, 740, 746
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Links

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Mathematica
    base=7; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Jan 30 2017 *)
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058903_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 6*l*n < 7**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(7),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,7)[1:]) == list(d):
                    break
            else:
                continue
            break
A058903_list = list(islice(A058903_gen(),20)) # Chai Wah Wu, May 10 2023

A058904 Inconsummate numbers in base 8: no number is this multiple of the sum of its digits (in base 8).

Original entry on oeis.org

42, 44, 51, 52, 60, 105, 109, 116, 124, 173, 177, 178, 181, 201, 205, 209, 210, 213, 214, 217, 233, 237, 241, 242, 245, 249, 250, 251, 254, 255, 269, 273, 277, 278, 282, 285, 287, 290, 298, 299, 300, 308, 336, 343, 348, 352, 397, 401, 402, 403
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Mathematica
    base=8; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Sep 21 2017 *)
  • Python
    from itertools import count, islice, combinations_with_replacement
def A058904_gen(startvalue=1): # generator of terms
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 7*l*n < 1<<3*(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(8),l):
                if (s:=sum(d)) > 0 and sorted(oct(s*n)[2:]) == list(map(str,d)):
                    break
            else:
                continue
            break
A058904_list = list(islice(A058904_gen(),20)) # Chai Wah Wu, May 09 2023

A058905 Inconsummate numbers in base 9: no number is this multiple of the sum of its digits (in base 9).

Original entry on oeis.org

46, 47, 48, 56, 58, 66, 76, 86, 136, 138, 167, 176, 222, 227, 228, 248, 258, 298, 302, 308, 312, 316, 318, 338, 343, 344, 347, 348, 352, 354, 356, 358, 362, 374, 383, 384, 392, 398, 402, 403, 404, 406, 407, 408, 411, 412, 414, 416, 422, 423
Offset: 1

Views

Author

N. J. A. Sloane, Jan 09 2001

Keywords

Crossrefs

Cf. A003635, A052491, A058898-A058907.

Programs

  • Maple
    For Maple code see A058906.
  • Python
    from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058905_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if l*n<<3 < 9**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(9),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,9)[1:]) == list(d):
                    break
            else:
                continue
            break
A058905_list = list(islice(A058905_gen(),20)) # Chai Wah Wu, May 10 2023

A277223 a(n) = A052489(n)/n.

Original entry on oeis.org

9, 9, 9, 12, 9, 9, 12, 9, 9, 9, 18, 9, 15, 9, 9, 18, 9, 9, 21, 9, 18, 18, 9, 9, 15, 18, 18, 21, 9, 9, 18, 18, 18, 12, 9, 18, 27, 18, 9, 12, 18, 18, 18, 18, 9, 21, 18, 18, 18, 9, 18, 18, 18, 18, 18, 9, 9, 15, 9, 9, 18, 0, 0, 17, 0, 18, 12, 9, 9, 12, 18, 18, 26, 27, 0
Offset: 1

Views

Author

Michel Marcus, Oct 06 2016

Keywords

Comments

a(n) is the largest multiplier k such that m = k*n is n times the sum of its decimal digits.
a(n) is never 1, 2, 3, 4, 5 or 6. Conjecture: if a(n) < 12 then a(n) = 0 or 9. - Robert Israel, Oct 06 2016

Examples

			a(2)=9 because m=2*9=18 is the largest m that is twice the sum of its decimal digits.
a(4)=12 because m=4*12=48 is the largest m that is four times the sum of its decimal digits.

Links

Crossrefs

Cf. A003635, A052489.

Programs

  • Maple
    N:= 200: # to get a(1) .. a(N)
A:= Vector(N):
for t from 1 while 9*(1+ilog10(t))*N >= t do
   k:= convert(convert(t,base,10),`+`);
   if t mod k = 0 and t <= N*k then
      A[t/k]:= max(A[t/k],k)
   fi
od:
convert(A,list); # Robert Israel, Oct 06 2016
  • Mathematica
    Table[Last[Select[Range[10^(IntegerLength@ n + 2)], n Total@ IntegerDigits@ # == # &] /. {} -> {0}]/n, {n, 75}] (* Michael De Vlieger, Oct 06 2016 *)
  • PARI
    a(n) = {nbd = 1; while (9*nbd*n > 10^nbd, nbd++); forstep(k=9*nbd*n, 1, -1, if (sumdigits(k)*n == k, return(k/n));); 0;}

Formula

a(n) = 0 for n in A003635.
a(n) = A007953(A052489(n)). - Altug Alkan, Oct 06 2016
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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